Arbitration and Stability in Cooperative Games with Overlapping Coalitions

Overlapping Coalition Formation (OCF) games (Chalkiadakis, Elkind, Markakis, Polukarov, & Jennings, 2010) are cooperative games where players can simultaneously participate in several coalitions. Capturing the notion of stability in OCF games is a difficult task: deviating players may continue to contribute resources to joint projects with non-deviators, and the crucial question is what payoffs the deviators expect to receive from such projects. Chalkiadakis et al. (2010) introduce three stability concepts for OCF games—the conservative core, the refined core, and the optimistic core—that are based on different answers to this question. In this paper, we propose a unified framework for the study of stability in the OCF setting, which encompasses the stability concepts considered by Chalkiadakis et al. (2010) as well as a wide variety of alternative stability concepts. Our approach is based on the notion of arbitration functions, which determine the payoff obtained by the deviators, given their deviation and the current allocation of resources. We provide a characterization of stable outcomes under arbitration. Furthermore, we propose a criterion for an outcome to be in the refined or optimistic core, complementing the results of Chalkiadakis et al. (2010) for the conservative core, and answering questions left open by Chalkiadakis et al. (2010). We also introduce and study a new notion of the core, namely, the sensitive core, which lies between the conservative core and the refined core. Additionally, we put forward criteria for nonemptiness of the arbitrated cores that extend the classic Bondareva–Shapley theorem. Finally, we show that OCF games with the conservative arbitration function are essentially equivalent to nonOCF games, by relating the conservative core of an OCF game to the core of a non-overlapping cooperative game, and use this result to obtain a strictly weaker sufficient condition for conservative core non-emptiness than the one given by Chalkiadakis et al. (2010).